In this paper, we consider a special case of a new type of mixed integer programming problem: the Transportation Network with Supply Eating Arcs. This problem arises from an application in intelligent [telecommunications] networks (IN). It contains within its definition many classical mixed integer programming problems. The problem is defined as a transportation network with an additional condition. For an arc to have positive flow, the node capacity of its originating node (i.e., a supply node) is decreased by a fixed amount depending on the arc. The special case that we consider is where this fixed amount is solely dependent on the demand node to which the arc is incident. In studying this problem, we are mainly interested in the feasibility of such problems and focus our attention on this aspect here. We discuss why finding a feasible solution is potentially difficult as well as various methods for finding one. In particular, we describe a set of solutions (in terms of solution structure) that always contains a feasible solution, if one exists. We then describe a method for finding a feasible solution based on enumerating this set. The main advantage of the solution procedure is that at no time do we need to explicitly solve an LP (or even a transportation problem).